%I A002425 M5036 N2174
%S A002425 1,1,1,17,31,691,5461,929569,3202291,221930581,4722116521,
%T A002425 968383680827,14717667114151,2093660879252671,86125672563201181,
%U A002425 129848163681107301953,868320396104950823611,209390615747646519456961
%N A002425 Denominator of Pi^(2n)/(GAMMA(2n)*(1-2^(-2n))*Zeta(2n)).
%C A002425 Consider the C(k)-summation process for divergent series: the series 
               Sum((-1)^n*(n+1)^k)==1-2^k+3^k-4^k+..., summable C(1) to the value 
               1/2 for k=0, is for each k>=1 exactly summable C(k+1) to the sum 
               s(k+1)=(2^(k+1)-1)*B(k+1)/(k+1) and so a(n)=Abs(numerator(s(2n))). 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 27 2002
%C A002425 Odd part of tangent numbers A000182 (even part is 2^A101921(n)). - Ralf 
               Stephan, Dec 21 2004
%C A002425 (-1)^n*a(n) is the numerator of Euler(2n+1,1). - njas, Nov 10 2009
%D A002425 H. Cohn, Bull. Am. Math. Soc., Sept. 1965, 681ff, esp. p. 688.
%D A002425 A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index 
               of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford 
               and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 73.
%D A002425 S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. 
               Math. 46 (1914), 33-51.
%D A002425 Konrad Knopp, Theory and application of infinite series, Divergent series, 
               Dover, p. 479
%D A002425 L. Oettinger, Archiv. Math. Phys., 26 (1856), see esp. p. 5.
%D A002425 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002425 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A002425 N. J. A. Sloane, <a href="b002425.txt">Table of n, a(n) for n = 1..300</
               a>
%H A002425 Konrad Knopp, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;
               c=umhistmath;idno=ACM1954.0001.001">Theorie und Anwendung der unendlichen 
               Reihen</a>, Berlin, J. Springer, 1922. (Original german edition of 
               "Theory and Application of Infinite Series")
%F A002425 a(n)=(-1)^n/n*(1-4^n)*B(2*n)*2^A001511(n) where B(k) denotes the k-th 
               Bernoulli number - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 
               30 2003
%F A002425 This is different from the sequence of numerators of the expansion of 
               cosec(x)-cot(x) - see A089171.
%F A002425 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 
               2009: (Start)
%F A002425 a(n) = denominator(4*n/((2^(2*n)-1)*bernoulli(2*n)))
%F A002425 (End)
%o A002425 (PARI) for(n=1,20,print1(abs(numerator(2*bernfrac(2*n)*(4^n-1)/(2*n))),
               ","))
%o A002425 (PARI) a(n)=if(n<1,0,(-1)^n/n*(1-4^n)*bernfrac(2*n)*2^valuation(2*n,2))
%Y A002425 Numerator given by A037239.
%Y A002425 Different from A089171.
%Y A002425 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 
               2009: (Start)
%Y A002425 Equals A160469(n)/A048896(n-1)
%Y A002425 Equals A089171(n)*A089170(n-1)
%Y A002425 (End)
%Y A002425 Sequence in context: A146667 A146462 A089171 this_sequence A046990 A059212 
               A058899
%Y A002425 Adjacent sequences: A002422 A002423 A002424 this_sequence A002426 A002427 
               A002428
%K A002425 nonn,frac,easy
%O A002425 1,4
%A A002425 N. J. A. Sloane (njas(AT)research.att.com).
%E A002425 The n=15 term was formerly incorrectly given as 86125672563301143.
%E A002425 Formula and cross-references edited by Johannes W. Meijer (meijgia(AT)hotmail.com), 
               May 21 2009